Title:
Almost all four-particle pure states are determined by their two-body marginals

Abstract: We show that generic pure states (states drawn according to the Haar measure)
of four particles of equal internal dimension are uniquely determined among all
other pure states by their two-body marginals. In fact, certain subsets of
three of the two-body marginals suffice for the characterization. We also
discuss generalizations of the statement to pure states of more particles,
showing that these are almost always determined among pure states by three of
their $(n-2)$-body marginals. Finally, we present special families of symmetric
pure four-particle states that share the same two-body marginals and are
therefore undetermined. These are four-qubit Dicke states in superposition with
generalized GHZ states.